One verifies that \(\widetilde{\Omega }_{\mathrm {r}}\)  is cyclic for \(\widetilde{\mathcal {M}}_{\mathrm {r},\varphi }^1\)  as above, hence we only have
to show that \(\widetilde{\mathcal {M}}_{\mathrm {r},\varphi }\)  and \(\widetilde{\mathcal {M}}_{\mathrm {r},\varphi }^1\)  commute. This has been done by Lemma REF .

One can actually show that \(\widetilde{\mathcal {M}}_{\mathrm {r},\varphi }^1 = (\widetilde{\mathcal {M}}_{\mathrm {r},\varphi })^{\prime }\) , so this confusing notation is justified.
Indeed, one has only to check the modular group of \(\widetilde{\mathcal {M}}_{\mathrm {r},\varphi }\)  with respect to \(\widetilde{\Omega }_{\mathrm {r}}\)  is
the same for the tensor product, which follows from the field representation in the next Section
and the argument of [1]}.
